Internal problem ID [8455]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 875.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {-y x -y+x^{5} \sqrt {y^{2}+x^{2}}-x^{4} \sqrt {y^{2}+x^{2}}\, y}{x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 79
dsolve(diff(y(x),x) = -(-x*y(x)-y(x)+x^5*(y(x)^2+x^2)^(1/2)-x^4*(y(x)^2+x^2)^(1/2)*y(x))/x/(x+1),y(x), singsol=all)
\[ \ln \left (\frac {2 x \left (\sqrt {2 x^{2}+2 y \relax (x )^{2}}+y \relax (x )+x \right )}{y \relax (x )-x}\right )+\frac {\sqrt {2}\, x^{4}}{4}-\frac {\sqrt {2}\, x^{3}}{3}+\frac {\sqrt {2}\, x^{2}}{2}-\sqrt {2}\, x -\ln \relax (x )+\sqrt {2}\, \ln \left (x +1\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 2.181 (sec). Leaf size: 150
DSolve[y'[x] == (y[x] + x*y[x] - x^5*Sqrt[x^2 + y[x]^2] + x^4*y[x]*Sqrt[x^2 + y[x]^2])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \tanh \left (\frac {3 x^4-4 x^3+6 x^2-12 x+12 \log (x+1)-25+12 c_1}{12 \sqrt {2}}\right ) \left (2+\sqrt {2} \tanh \left (\frac {3 x^4-4 x^3+6 x^2-12 x+12 \log (x+1)-25+12 c_1}{12 \sqrt {2}}\right )\right )}{\sqrt {2}+2 \tanh \left (\frac {3 x^4-4 x^3+6 x^2-12 x+12 \log (x+1)-25+12 c_1}{12 \sqrt {2}}\right )} \\ y(x)\to x \\ \end{align*}